So, are Hakka and Taiwanese "dialects" or languages in their own right?
Hakka, Taiwanese, Cantonese, Shanghaiese (Wu), and Mandarin are all different dialects of Chinese. They are different at branch level which means that are as different to each other as French to Spanish. But in written form they are all identical to each other (written and spoken Chinese could be very different except Mandarin).
What we call Taiwanese is identical to Hokkien spoken by most Chinese living in Malaysia and Singapore. (It is called such because Holos Chinese constitute a majority of in Taiwan) In theory it should be called Minnan language (Southern Fujian/Hokkien) but Taiwanese seems to have more cutting-edge words as it is the most developed part of China. The difference between "Hokkien" spoken in Singapore, Taiwanese spoken in Tainan, and Minnan spoken in Xiamen is like difference between American and New Zealand and British English.
Hakka is another Chinese dialect that is also spoken in Guangdong, Sichuan, Jiangxi, Guangxi - provinces also with lots of Hakka Chinese. It has no relation to Taiwanese/Hokkien/Minnan.